Linear relations in families of powers of elliptic curves

TL;DR

This paper proves finiteness results for special parameters where multiple points on a family of elliptic curves satisfy independent relations, contributing to the understanding of unlikely intersections in algebraic geometry.

Contribution

It establishes finiteness of parameters for which multiple linearly independent points on a family of elliptic curves satisfy independent relations, advancing conjectures on unlikely intersections.

Findings

Finiteness of such parameters is proven for the Legendre family.

Results relate to conjectures on unlikely intersections in abelian varieties.

Extends previous work on torsion points and relations on elliptic curves.

Abstract

Motivated by recent work of Masser and Zannier on simultaneous torsion on the Legendre elliptic curve $E_\lambda$ of equation $Y^2=X(X-1)(X-\lambda)$, we prove that, given $n$ linearly independent points $P_1(\lambda), ...,P_n(\lambda)$ on $E_\lambda$ with coordinates in $\bar{\mathbb{Q}(\lambda)}$, there are at most finitely many complex numbers $\lambda_0$ such that the points $P_1(\lambda_0), ...,P_n(\lambda_0)$ satisfy two independent relations on $E_{\lambda_0}$. This is a special case of conjectures about Unlikely Intersections on families of abelian varieties.